Chern--Simons Gravity from 3+1 Dimensional Gravity

TL;DR

This paper demonstrates how 3+1-dimensional gravity can be reduced to 2+1 and 1+1 dimensions within a gauge theoretical framework, linking solutions across different dimensions and including Chern-Simons gravity.

Contribution

It provides a gauge-theoretic dimensional reduction from 3+1 to lower-dimensional gravity, connecting known solutions like black holes and Liouville gravity across dimensions.

Findings

3+1D gravity reduces to 2+1D and 1+1D within Poincaré gauge theory.

2+1D solutions with negative cosmological constant map to 3+1D vacuum solutions.

The reduction includes Chern-Simons-Witten gravity and known black hole solutions.

Abstract

In the context of a Poincar\'e gauge theoretical formulation, pure gravity in 3+1-dimensions is dimensionally reduced to gravity in 2+1-dimensions with or without cosmological constant $\Lambda$. The dimensional reductions are consistent with the gauge symmetries, mapping ISO(3, 1) gauge transformations into ISO(2,1) ones. One of the reductions leads to Chern-Simons-Witten gravity. The solutions of 2+1-gravity with $\Lambda\le 0$ (in particular the black-hole solution recently found by Banados, Teitelboim and Zanelli) and those of 1+1-dimensional Liouville gravity, are thus mapped into 3+1-dimensional vacuum solutions.